Example 2: Rayleigh Method for Natural Period
Example 2: Rayleigh Method for Natural Period
Rayleigh Method as Defined in ASCE Section 15.4.4
There are cases where an engineer wants to use a quick hand calc method to determine or verify the
natural frequency of a structure. The Rayleigh method presented in ASCE ‐7 provides a relatively simple
method of doing this which can work nicely even for relatively complex structures.
Below is a 10 story steel frame structure which has gravity loading (or mass) at each defined floor level.
In additional it has been subjected to lateral wind forces at each level. The model was then solved in
RISA‐3D and the deflection at each floor level was determined.
Project Definition: Rayleigh Method.r3d
16
Example 2: Rayleigh Method for Natural Period
ASCE Equation 15.4‐6 gives the Rayleigh equation for natural period calculations as the following:
2
∑
∑
w i = seismic weight at level i
f i = Applied Lateral Force at level i
d i = Lateral Deflection at level i when subjected to applied forces.
This formula is very similar to our simplified single degree of freedom example where:
2
∆
2
The Rayleigh method just gives us a convenient way to extend this concept into multi degree of freedom
systems.
One thing interesting to note is that the applied lateral forces do not need to be earthquake forces. In
this case, we are using the wind loads. Below is an Excel spreadsheet summary which calculates these
values for us:
Floor
Level
Floor
Weight
(kips)
Wind
force
(kips)
Deflection due
to wind
(inches)
m_i*delta_i^2
f_i*delta_i
10
180
30.22
8.15
30.93
246.24
9
216
21.94
7.63
32.50
167.30
8
216
19.57
6.88
26.46
134.64
7
216
17.20
6.08
20.69
104.64
6
216
14.83
5.16
14.90
76.57
5
216
12.45
4.25
10.08
52.87
4
216
10.08
3.31
6.14
33.40
3
216
7.71
2.46
3.37
18.94
2
216
5.34
1.57
1.38
8.39
1
216
2.97
0.78
0.34
2.33
Totals
2124
146.80
845.31
17
Example 2: Rayleigh Method for Natural Period
Finishing off the calculation, we get:
2
146.8
845.31
2.62
We find that the Rayleigh method exactly matches exactly the value for the first mode that we get when
we solve for the mode shapes and frequencies in RISA‐3D as shown below.
We can take that procedure and use it for more advanced purposes like investigating the effect of 2nd
order or non‐linear effects on the natural period of our structure.
For example: We have already discussed how the RISA‐3D eigen solution does not automatically account
for the P‐Delta effect. Well, the Rayleigh method can be used to determine the approximate change in
natural frequency due to the softening effect of these second order effects. We do this merely by
including the P‐Delta effect in our static analysis. For this model, the results are shown below:
Floor
Level
Floor
Weight
(kips)
Wind
force
(kips)
Deflection due
to wind
(inches)
m_i*delta_i^2
f_i*delta_i
10
180
30.22
8.68
35.13
262.42
9
216
21.94
8.14
37.07
178.67
8
216
19.57
7.37
30.33
144.15
7
216
17.20
6.53
23.82
112.29
6
216
14.83
5.55
17.22
82.30
5
216
12.45
4.57
11.68
56.91
4
216
10.08
3.57
7.12
35.97
3
216
7.71
2.65
3.92
20.41
2
216
5.34
1.69
1.59
9.02
1
216
2.97
0.84
0.40
2.50
Totals
2124
168.27
904.63
Finishing off the calculation, we get:
18
Example 2: Rayleigh Method for Natural Period
2
168.27
904.63
2.71
The exercise is useful in that we can expect to see only about a 3% increase in our natural period
despite the fact that our static stiffness gets reduced by about 6%.
Notes:
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